A manifold $M$ is a second-countable, Hausdorff, locally Euclidean topological space. Obviously, there are advantages to requiring $M$ to be locally Euclidean, i.e. in some cases this allows $M$ to be endowed with a smooth structure.
The "locally Euclidean" axiom, outside of the setting it creates for the establishment of a possible calculus on $M$, is just a statement about the relationship $M$ has to a very particular metric space, namely $\mathbb{R}^n$. If one has no interest in doing calculus on $M$, I see no reason to restrict this "relationship" to $\mathbb{R}^n$. This motivates the following definition:
Let $A$ be a second-countable metric space. An $A$-manifold is a topological space $M$ such that $M$ is second-countable, Hausdorff and:
For each $p \in M$ there is an open set $U \subseteq M$ with $p \in U$, an open set $U^* \subseteq A$ and a homeomorphism $\phi: U \rightarrow U^*$.
Locally Euclidean manifolds then become the special case $A= \mathbb{R}^n$ for some $n$. One could of course generalize further, i.e. require $A$ only to be a topological space, but at that point the working mathematician is really only doing general topology.
To anyone's knowledge, have $A$-manifolds, or similar structures, been explored in detail? That is, is there any literature exploring the properties of these or similar structures?
NOTE: Also, if anyone has an objection to my statement: "If one has no interest in doing calculus on $M$, I see no reason to restrict this "relationship" to $\mathbb{R}^n$," please mention it.